Teaching a horse showmanship is very different for different horses. For pivot, I start just getting the horse to turn in small circles away from me, to just get the horse used to moving away from me as I walk into him. Then start tapping the horse on the shoulder with a lead or a crop to move the shoulder. Nothing about teaching showmanship is quick, it takes lots of practice and consitancy.
I start out by squaring the horse up. Once its squared up push the horse's shoulder over until it crosses over the legs. Once it does that pat him and say 'good boy!' Walk a couple of steps then stop, square up, then push into the shoulder. Once your horse gets the hang of you going into his shoulder and moving, start asking for more than one step. Ask for two, then three, then four, etc.
Make sure when your pivoting, your horse pivots on his RIGHT hind foot. If your horse pivots with his LEFT hind he will loose points.
If your horse pivots with his left hind, ask him to start pivoting. Once he pivots and he's about to move is RIGHT hind foot, slightly pull him forward so that his LEFT hind foot moves, rather than his RIGHT hind foot.
Sorry for the caps but some people get confused with this subject. (: